Let \(\mathcal {H}_X\)  be a Hilbert space equipped with an inner product \(\langle \cdot , \cdot \rangle \)  and an induced norm \(\Vert \cdot \Vert \) . For example, when the space \(\mathcal {H}_X = L_2(T)\)  is the set of all square integrable functions over a compact interval \(T\) , the inner product is \(\langle x, x^{\prime }\rangle = \int _T x(t) x^{\prime }(t)\,\mathrm {d}t\)  and the norm is \(\Vert x\Vert  = \big (\int _T x^2(t) \,\mathrm {d}t\big )^{1/2}\)  for \(x,x^{\prime }\in \mathcal {H}_X\) . Given a random element \(x\in \mathcal {H}_X\)  with zero mean and finite second moment \(\mathbb {E}\Vert x\Vert ^2\) , its covariance operator \(\mathcal {C}_{XX}\) : \(\mathcal {H}_X \mapsto \mathcal {H}_X\)  is a mapping such that \(u\in \mathcal {H}_X\)  is mapped to \(\mathbb {E}\langle u,x\rangle x\) . Equivalently, we can write \(\mathcal {C}_{XX}= \mathbb {E}\left( x\otimes x\right)\) , where \(\otimes \)  is tensor product such that \(\left(x\otimes x\right) u = \langle x, u\rangle \times x\)  for any \(u\in \mathcal {H}_X\) . Covariance operator generalizes the concept of covariance matrix in multivariate statistics.
For any \(u,v \in \mathcal {H}_X\) , it holds that \(\langle \mathcal {C}_{XX} u, v\rangle = \mathbb {E}(\langle x, u\rangle \langle x, v\rangle )\) . We can see that \(\langle \mathcal {C}_{XX} u, v\rangle \)  quantifies the correlation between the random scalars \(\langle x, u\rangle \)  and \(\langle x, v\rangle \) . A comprehensive introduction to these FDA concepts can be found in [1]}.
